How can I export and store light curves?
I would encourage you to start with the CDL node, as it is easy to replicate and generate looks that can be loaded with OpenColorIO.

If you are using a complex node chain to modify an image, you can reuse your crafted looks. It will require a bit of learning and patience. If you wish to dive into this subject, feel free to wade through this longwinded post written by an idiot. Also note, Blender's curves (and the entire UI) isn't colour managed by and as a result it is not likely responding the way that one might believe it is.

This post over at Blender Artists is quite useful as well.